You’ve operated with fractions and decimals, prefer 3.8 and
. This numbers can be found between the integer numbers on a number line. Over there are various other numbers that can be uncovered on a number line, too. As soon as you include all the number that can be placed on a number line, you have the actual number line. Let"s destruction deeper into the number line and see what those numbers look like. Let’s take a closer look at to watch where this numbers fall on the number line.You are watching: Name all sets to which each number belongs
The portion , mixed number
, and decimal 5.33… (or ) all represent the same number. This number belongs come a set of numbers that mathematicians contact rational numbers. Reasonable numbers space numbers that can be composed as a proportion of two integers. Nevertheless of the form used, is rational since this number can be composed as the ratio of 16 end 3, or .Examples of rational numbers incorporate the following.
0.5, as it can be composed as
, as it deserve to be written as−1.6, as it deserve to be composed as
4, together it have the right to be written as
10, as it deserve to be written as
All of these numbers can be written as the ratio of 2 integers.
You can locate these points ~ above the number line.
In the following illustration, points are displayed for 0.5 or , and also for 2.75 or
.As you have actually seen, rational numbers deserve to be negative. Each hopeful rational number has actually an opposite. Opposing of is
, for example.Be cautious when put negative numbers ~ above a number line. The an unfavorable sign means the number is come the left that 0, and also the absolute worth of the number is the distance from 0. For this reason to place −1.6 on a number line, you would uncover a suggest that is −1.6 or 1.6 systems to the left of 0. This is much more than 1 unit away, but less 보다 2.
Example  
Problem  Place on a number line. 
It"s helpful to an initial write this improper fraction as a combined number: 23 divided by 5 is 4 v a remainder of 3, so is .  
Since the number is negative, you can think of it as relocating units come the left that 0. will be between −4 and also −5.  
Answer 
Which the the following points to represent ? Show/Hide Answer A) Incorrect. This allude is simply over 2 units to the left of 0. The point should it is in 1.25 devices to the left that 0. The correct answer is suggest B. B) Correct. An adverse numbers are to the left the 0, and should be 1.25 units to the left. Allude B is the only suggest that’s an ext than 1 unit and less than 2 systems to the left that 0. C) Incorrect. An alert that this point is between 0 and also the very first unit note to the left the 0, for this reason it represents a number in between −1 and 0. The point for should it is in 1.25 units to the left of 0. You may have correctly uncovered 1 unit to the left, however instead of proceeding to the left another 0.25 unit, you moved right. The correct answer is allude B. D) Incorrect. Negative numbers are to the left of 0, no to the right. The suggest for should it is in 1.25 devices to the left that 0. The exactly answer is point B. E) Incorrect. This point is 1.25 units to appropriate of 0, for this reason it has actually the correct distance yet in the not correct direction. An adverse numbers are to the left of 0. The correct answer is allude B. Comparing rational Numbers When 2 whole numbers room graphed top top a number line, the number to the appropriate on the number line is constantly greater 보다 the number top top the left. The exact same is true when comparing 2 integers or rational numbers. The number come the ideal on the number line is always greater than the one ~ above the left. Here space some examples.
Irrational and also Real Numbers There are also numbers that space not rational. Irrational numbers cannot be composed as the proportion of 2 integers. Any square root of a number that is not a perfect square, for instance , is irrational. Irrational numbers space most frequently written in among three ways: as a source (such together a square root), using a unique symbol (such together ), or as a nonrepeating, nonterminating decimal. Numbers with a decimal part can either be terminating decimals or nonterminating decimals. Terminating method the number stop at some point (although you can always write 0s at the end). Because that example, 1.3 is terminating, because there’s a critical digit. The decimal type of is 0.25. Terminating decimals are always rational. Nonterminating decimals have actually digits (other than 0) that proceed forever. Because that example, take into consideration the decimal form of , which is 0.3333…. The 3s continue indefinitely. Or the decimal kind of , i m sorry is 0.090909…: the succession “09” proceeds forever.In addition to being nonterminating, these two numbers are additionally repeating decimals. Your decimal parts are make of a number or sequence of numbers that repeats again and also again. A nonrepeating decimal has digits that never form a repeating pattern. The worth of, for example, is 1.414213562…. No issue how much you lug out the numbers, the number will never ever repeat a ahead sequence. If a number is terminating or repeating, it must be rational; if the is both nonterminating and nonrepeating, the number is irrational.
The set that real numbers is do by combine the collection of reasonable numbers and also the set of irrational numbers. The actual numbers encompass natural numbers or counting numbers, entirety numbers, integers, rational number (fractions and also repeating or end decimals), and also irrational numbers. The set of actual numbers is all the number that have a place on the number line.
